Τοπολογική Ίνα
Τοπολογική Ίνα fiber thumb|300px| [[Τοπολογική Ίνα ]] thumb|300px| [[Αντίστροφη Συνάρτηση ]] - Ένα Τοπολογικό Δόμημα Ετυμολογία Η ονομασία "τοπολογική" σχετίζεται ετυμολογικά με την λέξη "τοπολογία". In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context: # In naive set theory, the fiber of the element y'' in the set ''Y under a map f'' : ''X → Y'' is the inverse image of the singleton \{y\} under ''f. # In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed. Ορισμοί Fiber in naive set theory Let f'' : ''X → Y'' be a map. The '''fiber' of an element y \in Y , commonly denoted by f^{-1}(y) , is defined as : f^{-1}(\{y\})=\{x \in X \, | \, f(x) = y\}. In various applications, this is also called: :* the inverse image of \{y\} under the map f'' :* the preimage of \{y\} under the map ''f :* the level set of the function f'' at the point ''y. The term level set is only used if f'' maps into the real numbers and so ''y is simply a number. If f'' is a continuous function and if ''y is in the image of f'', then the 'level set' of ''y under f'' is a curve in 2D, a surface in 3D, and more generally a hypersurface of dimension ''d-1. Fiber in algebraic geometry In algebraic geometry, if f'' : ''X → Y'' is a morphism of schemes, the '''fiber' of a point p'' in ''Y is the fibered product X\times_Y \mathrm{Spec}\, k(p) where k''(''p) is the residue field at p''. Terminological variance The recommended practice is to use the terms ''fiber, inverse image, preimage, and level set as follows: :* the fiber of the element y'' under the map ''f :* the inverse image of the set \{y\} under the map f'' :* the ''preimage of the set \{y\} under the map f'' :* the ''level set of the function f at the point y''. By abuse of terminology, the following terminology is sometimes used but should be avoided: :* the ''fiber of the map f'' at the element ''y :* the inverse image of the map f'' at the element ''y :* the preimage of the map f'' at the element ''y :* the level set of the point y'' under the map ''f. Ανάλυση If A'' is any subset of the domain ''X, then f''(''A) is the subset of the codomain Y'' consisting of all images of elements of A. We say the ''f(A'') is the ''image of A under f. The image of f'' is given by ''f(X''). On the other hand, the ''inverse image (or preimage, complete inverse image) of a subset B'' of the codomain ''Y under a function f'' is the subset of the domain ''X defined by: : f^{-1}(B) = \{x \in X : f(x) \in B\}. So, for example, the preimage of {4, 9} under the squaring function is the set {−3,−2,2,3}. The term range usually refers to the image, but sometimes it refers to the codomain. By definition of a function, the image of an element x'' of the domain is always a single element ''y of the codomain. Conversely, though, the preimage of a singleton set (a set with exactly one element) may in general contain any number of elements. For example, if f''(''x) = 7 (the constant function taking value 7), then the preimage of {5} is the empty set but the preimage of {7} is the entire domain. It is customary to write f''−1(''b) instead of f''−1({''b}), i.e. : f^{-1}(b) = \{x \in X : f(x) = b\}. This set is sometimes called the fiber of b'' under ''f. Use of f''(''A) to denote the image of a subset A'' ⊆ ''X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f''[''A] for the set { f''(''x): x ∈ A'' }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write ''f−1[B''] and ''f−1[b] for the preimage of a set and a singleton. Υποσημειώσεις Εσωτερική Αρθρογραφία * Συνάρτηση * Ινοδέσμη * Πολύπτυχο * Fibration * Fiber bundle * Fiber product * Image (category theory) * Image (mathematics) * Inverse relation * Kernel (mathematics) * Level set * Preimage * Relation * Zero set Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *Introduction to Bundles *[ ] Κατηγορία:Τοπολογικά Δομήματα